A classical example of a mathematical model for oscillations in a biological system is the Selkov oscillator, which is a simple description of glycolysis. It is a system of two ordinary differential equations which, when expressed in dimensionless variables, depends on two parameters. Surprisingly it appears that no complete rigorous analysis of the dynamics of this model has ever been given. In this paper several properties of the dynamics of solutions of the model are established. With a view to studying unbounded solutions a thorough analysis of the Poincaré compactification of the system is given. It is proved that for any values of the parameters there are solutions which tend to infinity at late times and are eventually monotone. It is shown that when the unique steady state is stable any bounded solution converges to the steady state at late times. When the steady state is unstable it is shown that for given values of the parameters either there is a unique periodic solution to which all bounded solutions other than the steady state converge at late times or there is no periodic solution and all solutions other than the steady state are unbounded. In the latter case each unbounded solution which tends to infinity is eventually monotone and each unbounded solution which does not tend to infinity has the property that each variable takes on arbitrarily large and small values at arbitrarily late times.

The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This is done with the help of a Poincaré compactification of the system and a shooting argument. This system was originally derived from another system with Michaelis–Menten kinetics. A Poincaré compactification of the latter system is carried out and this is used to show that the Michaelis–Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models for the same biological system in the literature.

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